Transport Phenomena.pdf

406 Chapter 12 Temperature Distributions with More Than One Independent Variable
12D.7. Asymptotic solution for small z for laminar tube flow with constant heat flux. Fill in the
missing steps between Eq. 12.2-23 and Eq. 12.2-24. Insertion of the expression for ф into Eq.
12.2-23 gives
в =
J
x LI (3) J x
Why do we introduce the symbols J and ~x>Next, exchange the order of integration to get
@ = ^9X Г UL f * Jexpf-pW*) \dx
J
X [Л\з) J x J
(12D.7-2)
Then perform the integration over x and obtain
в(г>' A) = Щ-1 ex P(~^3) ~ Ц J o
" ^ exp(-^3) ^ - ^ x exp(-^3
) rf^J I (12D.7-3)
Then use the definitions T(a) = f% 1 а ~^е~ 1 &\ and T(a, x) = /J F~ A e~ [ dt for the complete and incomplete
gamma functions.
12D.8. Forced conduction heat transfer from a flat plate (thermal boundary layer extends beyond
the momentum boundary layer). Show that the result analogous to Eq. 12.4-14 for A > 1 is 13
3
H. Schlichting, Boundary-Layer Theory, 7th edition, McGraw-Hill, New York (1979), p. 306.